Basis Lab

Answers · The physics underneath

Why does a spinning book suddenly start tumbling instead of staying on its axis?

In shortSpinning a book about its middle-sized axis is unstable: the conserved quantities still allow it to turn over, and any tiny wobble grows until it flips.

Set a closed book spinning freely about the axis across its width, the middle of its three resistances, and it holds for a moment and then turns over. Nothing has been lost while that happened. The book keeps its energy of rotation and the size of its angular momentum exactly, from the throw to the catch. Those two laws confine the motion to a curve, and the curve through a spin about the middle axis runs all the way to the far side. The flip was in the initial conditions from the start; all a nudge does is choose when.

2E=I1ω12+I2ω22+I3ω32L2=I12ω12+I22ω22+I32ω322E = I_1\omega_1^2 + I_2\omega_2^2 + I_3\omega_3^2 \qquad L^2 = I_1^2\omega_1^2 + I_2^2\omega_2^2 + I_3^2\omega_3^2

Take the three resistances to be 1, 2 and 3, easiest first, and spin the book about the middle one at rate 1. Give it a small nudge about each of the other two, say 0.02. Then the two conserved sums start at

2E=1(0.02)2+2(1)2+3(0.02)2=0.0004+2+0.0012=2.00162E = 1(0.02)^2 + 2(1)^2 + 3(0.02)^2 = 0.0004 + 2 + 0.0012 = 2.0016
L2=1(0.02)2+4(1)2+9(0.02)2=0.0004+4+0.0036=4.004L^2 = 1(0.02)^2 + 4(1)^2 + 9(0.02)^2 = 0.0004 + 4 + 0.0036 = 4.004

Now ask what the book can be doing when the middle-axis spin has fallen to zero, which is the moment it looks stopped and about to reverse. Put q=0q = 0 and let the other two rates be pp and rr. The two conservation laws give two equations in the two remaining squares, so the state is pinned rather than free.

p2+3r2=2.0016p2+9r2=4.004p^2 + 3r^2 = 2.0016 \qquad p^2 + 9r^2 = 4.004
6r2=2.0024  ⇒  r2=0.33373,  r=0.5777p2=2.0016−3(0.33373)=1.0004,  p=1.00026r^2 = 2.0024 \;\Rightarrow\; r^2 = 0.33373,\; r = 0.5777 \qquad p^2 = 2.0016 - 3(0.33373) = 1.0004,\; p = 1.0002

The middle-axis spin is gone, and in its place the book is spinning about its long axis at 1.0, with a smaller rate about the covers. Substitute back and the two sums are still 2.0016 and 4.004. That is a book turned over: a state with the same energy and the same angular momentum as the one it was thrown in, reached without anything external acting. The rate about the long axis grew fiftyfold from 0.02 to 1.0002, and fifty is exactly how far the nudge has to be amplified for the middle spin to be cancelled.

Why the guess fails

The natural expectation is that conserved means steady: if energy and angular momentum are fixed, the rotation ought to stay where it was put. Conservation constrains where a system may go; it makes no promise that the place is near the start. Two conditions on three numbers leave a one-dimensional set, and a curve through the starting point can loop back close to it or run right across to its opposite. There is a second guess worth naming, that the flip is caused by the clumsiness of the throw or by air. Reduce the nudge and the flip arrives later, not never: a hundredfold smaller nudge makes the same excursion, because the excursion is the curve and the nudge only sets the speed along it.

FIG 1How fast a small deviation grows about an axis
the growth rate of a deviation; real means doubling and doubling again, imaginary means it circles instead of growing
the initial spin rate, so throwing harder brings the flip sooner, in proportion
the gap between the middle and easiest resistances, positive here because 2 is bigger than 1
the gap between the hardest and middle resistances, positive here because 3 is bigger than 2
Written for a spin about axis 2. The product under the root is positive for exactly one axis, the middle one, and that is the whole distinction between a spin that holds and a spin that turns over.

With 1, 2, 3 the root is (1)(1)/(1×3)=1/3=0.5774\sqrt{(1)(1)/(1 \times 3)} = \sqrt{1/3} = 0.5774 per unit time, and the doubling time for a deviation is ln⁡2/0.5774=1.20\ln 2 / 0.5774 = 1.20. Growing from 0.02 to 1 needs a factor of 50, so ln⁡50/0.5774=6.8\ln 50 / 0.5774 = 6.8 time units. From 0.0002 to 1 needs a factor of 5000, so ln⁡5000/0.5774=14.8\ln 5000 / 0.5774 = 14.8, a little over twice as long. That factor of two is why no throw is careful enough: shrinking the error costs only a logarithmic delay.

FIG 2
axis spun aboutproduct of the two bracketswhat a nudge does
1, easiest (down the spine)circles, never grows; the spin holds
2, middle (across the width)doubles every 1.20 units; the book turns over
3, hardest (through the covers)circles, never grows; the spin holds

Where it stops applying

The treacherous axis exists only because the three resistances are all different. Make two of them equal and the bracket for the third goes to zero, so the growth rate vanishes and every axis is neutral; make all three equal, as for a sphere or a cube, and no axis is distinguished at all. The other assumption is that nothing is holding the body. A hinge, a string or a track applies a torque, angular momentum is then no longer fixed, and a spin about the middle axis can be kept upright indefinitely, which is how a propeller shaft is balanced. It is the freedom of the body, not its shape, that gives the flip its inevitability.

Worth keeping

  • A free rigid body conserves both its rotational energy and the squared magnitude of its angular momentum, and those two constraints together fix a one-dimensional curve of allowed states.
  • About the largest and smallest principal axes that curve is a small closed loop; about the middle axis it crosses to the opposite spin, so the body turns over.
  • The deviation grows like a real exponential, so shrinking the initial wobble only delays the flip by a logarithm, never prevents it.

Where this goes next

This question is one step of Motion and What Is Conserved, a course that works through its subject from the beginning. The page that teaches this part of it is Why a Spinning Book Tumbles, and it reads the whole way through whether or not you have an account.

What an account changes is the practice: problems standing under every idea, marked with a reason rather than a tick, in a copy of the course that is yours.

Read that pageThe whole course

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